This disclosure deals with systems and methods using a Hermetic Transform, as well as related transforms, for applications such as directional or non-directional reception and/or transmission of signals, which use of phased-array devices and systems and/or waveforms that utilize Hermetic Transforms to create and decipher a plurality of modulated sub-carrier tones used in data transmission. For directional transmission, the Hermetic Transform (and related transforms) can be designed using an array manifold, effectively a complex calibration response vectors from the array in question to signal arrivals from different directions, whether developed from a mathematical model or from collected data, arranged in a particular fashion. The transform can be utilized for receiver and/or transmit beams to provide narrower main-lobes than classical methods would typically allow. Further background about the Hermetic Transform can be found in U.S. Pat. No. 8,064,408, incorporated herein by reference in its entirety and for all purposes.
For either directional or non-directional data transmission, Orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple sub-carrier frequencies, where the sub-carriers are selected so as to be non-interfering (orthogonal). OFDM and OFDMA have been adopted for wideband digital communication, both wireless over copper wires, used in applications such as digital television and audio broadcasting, DSL broadband internet access, wireless networks, such as 802.11 and in 4G mobile communications.
In traditional OFDM, a number of closely spaced orthogonal sub-carrier signals are used to carry data. The data is divided into several parallel data streams or channels, one for each sub-carrier. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation (QAM), or phase-shift keying, (PSK) at a low symbol rate, maintaining total data rates similar to conventional single-carrier modulation schemes in the same bandwidth. Advantages of OFDM vis a vis single-carrier schemes include the ability to cope with severe channel conditions. Channel equalization is simplified because OFDM may be viewed as using many slowly modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate intersymbol interference (ISI) and utilize echoes and time-spreading from multipath propagation in order to achieve diversity gain, for signal-to-noise ratio improvement.
With standard OFDM, complex modulations are passed through an Inverse Fast Fourier Transform (IFFT) which produces real-part or Q-waveform and imaginary-part or I-waveform components of the signal. These real and imaginary parts are impressed onto cosine and sine waveforms (sub-carriers) and summed to provide a real signal for transmission, which in the case of an application such as Wi-Fi (for example, 802.11n) is mixed onto a radio-frequency carrier, e.g. at 2.4 or 5.8 GHz. Demodulation is accomplished in an inverse fashion of the modulation, using a Fast-Fourier Transform.
A fundamental limit on the data-rate that can be carried by the signal is Shannon's channel capacity, C=B log2(1+S/N), where C is the channel capacity in bits/second, B is the signal bandwidth, and S/N is the ratio of signal power to noise power. Another limitation of OFDM is the required spacing to achieve orthogonality between sub-carriers using FFT/IFFT. Two factors would improve the data-rate of OFDM, one being the reduction of noise (increasing S/N) and the other being achieving orthogonality with more closely spaced sub-carriers, thus carrying more data in the parallel sub-carrier channels.